Bounds for the Boxicity of Mycielski Graphs 1

Volume 13, Number 1, Pages 63–78 ISSN 1715-0868

BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS

AKIRA KAMIBEPPU

Abstract. A box in Euclidean k-space is the Cartesian product I1 × I2 Ik, where Ij is a closed interval on the real line. The boxicity × · · · × of a graph G, denoted by box(G), is the minimum nonnegative integer k such that G can be isomorphic to the intersection graph of a family of boxes in Euclidean k-space. Mycielski [11] introduced an interesting graph operation that extends a graph G to a new graph M(G), called the Mycielski graph of G. In this paper, we observe the behavior of the boxicity of Mycielski graphs. The inequality box(M(G)) box(G) holds for a graph G, and hence ≥ we are interested in whether the boxicity of the Mycielski graph of G is more than that of G or not. Here we give bounds for the boxicity of Mycielski graphs: for a graph G with l universal vertices, the inequalities box(G) + l/2 box(M(G)) θ(G) + l/2 + 1 hold, where θ(G) is d e ≤ ≤ d e the edge clique cover number of the complement G. Further observations determine the boxicity of the Mycielski graph M(G) if G has no universal vertices or odd universal vertices and satisﬁes box(G) = θ(G). We also present relations between the Mycielski graph M(G) and its generalizations M3(G) and Mr(G) in the context of boxicity, which will encourage us to calculate the boxicity of M(G) and M3(G).

1. Introduction The notion of boxicity of graphs was introduced by Roberts [13]. It has applications in some research ﬁelds, like niche overlap in ecology (see [14, 15]) and ﬂeet maintenance in operations research (see [12]). Roberts [13] proved that the maximum boxicity of graphs with n vertices is n/2 (also see [7]), where x denotes the largest integer at most x. Cozzensb c [6] proved that the taskb ofc computing boxicity of graphs is NP-hard. Some researchers have attempted to calculate or bound boxicity of graphs with special struc- ture. Roberts [13] showed that the boxicity of a complete k-partite graph

Kn1,n2,...,nk is the number of ni which is at least 2. Scheinerman [16] proved that the boxicity of outer planar graphs is at most 2. Thomassen [17] proved

Received by the editors August 5, 2015, and in revised form April 27, 2017. 2000 Mathematics Subject Classiﬁcation. 05C62, 05C76. Key words and phrases. boxicity; chromatic number; cointerval graph; edge clique cover number; Mycielski graph. This work was supported by Grant-in-Aid for Young Scientists (B), No.25800091.

c 2018 University of Calgary

63 64 AKIRA KAMIBEPPU that the boxicity of planar graphs is at most 3. Cozzens and Roberts [7] in- vestigated the boxicity of split graphs. As Chandran et al. [5] say, not much is known about boxicity of most of the well-known graph classes. They proved that the boxicity of a graph G is at most tw(G) + 2, where tw(G) is the treewidth of G, and presented upper bounds for chordal graphs, circu- lar arc graphs, AT-free graphs, co-comparability graphs, and permutation graphs. Recently, Chandran et al. [1] found the following relation between boxicity and chromatic number. Theorem 1.1 ([1], Theorem 6.1). Let G be a graph with n vertices. If box(G) = n/2 s for s 0, the inequality χ(G) n/(2s + 2) holds, where χ(G) is the chromatic− number≥ of G. ≥ Theorem 1.1 implies that, if the boxicity of a graph with n vertices is very close to the maximum boxicity n/2 , the chromatic number of the graph must be very large. The converseb doesc not hold in general. The complete graph Kn is an example of a graph whose boxicity is small, even though its chromatic number is large. Also there are bipartite graphs with arbitrary large boxicity (see Section 5.1 in [1] and also see [2]). However, a graph operation increasing chromatic number may admit increasing boxicity. For example, the join of two graphs, taking the disjoint union of two graphs and adding all edges between them is desired one. The behavior of boxicity has been studied in the context of various graph operations (see [3, 4, 18] for example). This paper is another attempt in this direction that studies the behavior of boxicity in the context of Mycielski’s graph operation. One purpose of this paper is to consider whether the behavior of boxicity is similar to that of chromatic number under Mycielski’s graph operation. Mycielski [11] found an interesting graph operation that extends a graph G to a new graph M(G), called the Mycielski graph of G or the Mycielskian of G. It is well-known that the chromatic number of the Mycielski graph of G is more than that of G, in fact, χ(M(G)) = χ(G) + 1 holds. We can construct (triangle-free) graphs with arbitrarily large chromatic numbers by using the graph operation. Here we present the definition of the graph M(G). Let V (G)i be a copy of the vertex set V (G) of a graph G, where i 1, 2 . For each vertex v V (G), the symbol vi denotes the vertex ∈ { } ∈ in V (G)i corresponding to v. The vertex set of M(G) is defined to be z V (G)1 V (G)2, the disjoint union of the set of a single new vertex z { } ∪ ∪ and copies V (G)1 and V (G)2. The edge set of M(G) is defined to be the union E1 E2 E3, where ∪ ∪ E1 = u1v1 uv E(G) ,E2 = u1v2, v1u2 uv E(G) , and { | ∈ } { | ∈ } E3 = zu2 u V (G) { | ∈ } and E(G) denotes the edge set of G (see Figure 1 for example). Note that the inequality box(M(G)) box(G) holds for a graph G since M(G) ≥ contains the subgraph induced by V (G)1, isomorphic to G. So, first we are interested in whether the boxicity of the Mycielski graph M(G) is strictly BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 65 greater than G, the same as the behavior of the chromatic number under the graph operation, as mentioned at the beginning of this paragraph. Many researchers have studied Mycielski graphs and have compared a graph G with M(G) under various graph invariants (see [8, 10] for example).

M(C4) M(C4) u2 u1 u2 u1

v v1 v2 2 v1 z z x1 x2 x2 x1

y2 y1 y2 y1

Figure 1. The Mycielski graph M(C4) of a cycle C4 and its complement M(C4).

In Section 3, we improve the trivial lower bound for the boxicity of the Mycielskian of a graph G in terms of the number of universal vertices of G. This implies that the boxicity of the Mycielski graph M(G) is more than that of G if the graph G has universal vertices. Also note that there is a graph G without universal vertices such that the boxicity of the Mycielski graph M(G) is more than that of G. While such examples of graphs appear, there is also a graph G such that box(M(G)) = box(G). As a conclusion, the behavior of boxicity is not similar to that of chromatic number under Mycielski’s graph operation in general. This leads to our next goal: Classify as many graphs as possible into box(M(G)) > box(G) or box(M(G)) = box(G). In Section 4, we discuss upper bounds for the boxicity of Mycielski graphs. Chandran et al. [1] proved that the inequality box(G) t(G)/2 + 1 holds for a graph G, where t(G) is the minimum cardinality of≤ a b vertexc cover of G. It is easy to see that t(M(G)) 2t(G) + 1 for a graph G, and hence we have box(M(G)) t(M(G))/2 +≤ 1 t(G) + 1. Here we present another upper bound for the≤ boxicityb of thec Mycielskian≤ of a graph G in terms of the edge clique cover number θ(G) of the complement G. We also consider graphs that satisfy the equality box(G) = θ(G). The family of graphs satisfying box(G) = θ(G) contains complete multipartite graphs, for example. Other examples of such graphs appear at the end of Section 4. As a result, our observations determine the boxicity of their Mycielski graphs if the original graphs have no universal vertices or odd universal vertices. In Section 5, we consider relations between the Mycielski graph and its generalization Mr(G), called the generalized Mycielski graph of G, in the context of boxicity, where r 3. We present upper bounds for the boxicity ≥ 66 AKIRA KAMIBEPPU of the generalized Mycielski graph Mr(G) in terms of M(G) for a bipartite graph G or in terms of M3(G) for a graph G. These results provide motivation for calculating the boxicity of M(G) and M3(G).

2. Preliminaries In this paper, all graphs are finite, simple and undirected. We use V (G) for the vertex set of a graph G. We use E(G) for the edge set of a graph G. An edge of a graph with endpoints u and v is denoted by uv. A vertex v of G is said to be universal if v is adjacent to all vertices in V (G) v . A graph is said to be trivial if E(G) is empty. For a subset V of V (G),\{ let} G V be the subgraph induced by V (G) V . For a subset E of E(G), let G − E be the subgraph on V (G) with E(G\) E as its edge set. A subset of V (G−) that induces a complete subgraph of G\is called a clique of G. For a graph G, its complement is denoted by G. The intersection graph of a nonempty family of sets is the graph whose vertex set is and F1 is adjacent to F2 if and F F only if F1 F2 = for F1, F2 . The intersection graph of a family of closed intervals∩ 6 on∅ the real line∈ is F called an interval graph. A graph G can be represented as the intersection graph of a family if there is a bijection between V (G) and such that two vertices of G areF adjacent if and only if the corresponding setsF in have nonempty intersection. A box in Euclidean F k-space is the Cartesian product I1 I2 Ik, where Ij is a closed interval on the real line. The boxicity× of× a graph· · · × G, denoted by box(G), is the minimum nonnegative integer k such that G can be represented as (isomorphic to) the intersection graph of a family of boxes in Euclidean k- space. The boxicity of a complete graph is defined to be 0. If G is an interval graph, box(G) 1. If H is an induced subgraph of G, box(H) box(G) holds by the definition.≤ ≤ A graph is a cointerval graph if its complement is an interval graph. Lekkerkerker and Boland [9] presented the forbidden subgraph characterization of interval or cointerval graphs. Cointerval graphs do not contain the complement of a cycle of length at least 4 as an induced subgraph, for example. It is easy to see that the union of a cointerval graph and isolated vertices is also a cointerval graph. A cointerval edge covering of a graph G is a family of cointerval spanning subgraphs of G such that each edge of G is in some graphC of . For a set X, the cardinality of X is denoted by X . The following theoremC is useful to calculate of the boxicity of graphs.| | Theorem 2.1 ([7], Theorem 3). Let G be a graph. Then, box(G) k if and only if there is a cointerval edge covering of G with = k. ≤ C | C | 3. A lower bound for the boxicity of Mycielski graphs

For a complete graph Kn, it is easy to see that box(M(Kn)) 1 > 0 = ≥ box(Kn) since M(Kn) is not complete by the deﬁnition of Mycielski graphs. We determine the boxicity of M(Kn) in the next section (see Lemma 4.1). BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 67

First we consider if the boxicity of the Mycielski graph of a graph G is more than that of G in general. Question 3.1. For a graph G, does the inequality box(M(G)) > box(G) hold? The following example shows that there exists a graph G such that the equality box(M(G)) = box(G) holds. Here Cn denotes a cycle with n vertices.

Example 3.2. The boxicity of the Mycielski graph of a cycle C4 is equal to 2. To check this, we give a cointerval edge covering of the complement M(C4) (see Figure 1). Let H1 and H2 be the graphs in Figure 2. Both graphs are cointerval spanning subgraphs of M(C4). Note that the disjoint union of a cointerval

u2 u1 u2 u1

v v1 v2 2 v1 z z

x2 x1 x2 x1 H1 H2 y2 y1 y2 y1

u1 u2 Iy2 v2 z Ix1 Iu1 Iz

x1 x2 Iv2 Ix2 Iu2

H v ,y y2 1 −{ 1 1}

Figure 2. Cointerval spanning subgraphs H1 and H2 of M(C4) and an interval representation of H1 v1, y1 . − { } graph and isolated vertices is also cointerval since these isolated vertices become pairwise adjacent universal vertices in the complement. Hence, it suﬃces to prove that H1 v1, y1 and H2 u1, x1 are cointerval, re- spectively. A family of intervals− { on} the real− line { with} intersection graph isomorphic to H1 v1, y1 can be found as in the bottom of Figure 2. Sim- − { } ilar arguments work for H2 u1, x1 . Also see that H1 and H2 cover all − { } edges of M(C4). The family H1,H2 is a desired cointerval edge covering { } of M(C4), and hence, box(M(C4)) 2 by Theorem 2.1. Also note that ≤ box(M(C4)) box(C4) = 2. ≥ 68 AKIRA KAMIBEPPU

Question 3.3. Is there a graph G such that the inequality box(M(G)) > box(G) holds? The distance between two vertices u and v in a graph G is defined by length of the shortest path from u to v in G and is denoted by dG(u, v). If there exist no paths from u to v in G, define dG(u, v) = . Let H1 and H2 be subgraphs ∞ of G. The distance between two subgraphs H1 and H2 in G, denoted by dG(H1,H2), is defined to be the minimum distance min dG(v1, v2) v1 { | ∈ V (H1), v2 V (H2) . The following lemma is a generalization of Corollary 3.6 in [7]. ∈ }

Lemma 3.4. Let G be a graph and H1, H2 induced subgraphs of the complement G. If d (H1,H2) 2, the following inequality holds: G ≥ box(G) box(H1) + box(H2). ≥ Proof. If either H1 or H2 is trivial, say H1, then H1 is complete. Hence, box(H1) = 0. Since H2 is an induced subgraph of G, we see that

box(G) box(H2) = box(H1) + box(H2) ≥ holds. In what follows, we may assume that H1 and H2 are nontrivial. The assumption d (H1,H2) 2 means that d (v1, v2) 2 for any vertex G ≥ G ≥ v1 of H1 and v2 of H2. Hence, an edge of H1 and an edge of H2 form 2K2, the disjoint union of two edges, as an induced subgraph of G. Moreover, we claim the following.

Claim 1. No cointerval spanning subgraphs of G contain an edge of H1 and an edge of H2.

Claim 2. We need at least box(Hi) cointerval spanning subgraphs of G to cover all edges of Hi, where i 1, 2 . ∈ { } Claim 1 follows from the forbidden subgraph characterization of cointerval graphs. In fact, cointerval graphs do not contain 2K2 as an induced subgraph. Claim 2 follows from Theorem 2.1. A cointerval subgraph of G with edges of H1 does not contain edges of H2. Thus, the inequality box(G) box(H1) + box(H2) holds. ≥ We can derive a positive answer to Question 3.3 by using Lemma 3.4. The following lemma is useful to make our answer more precise. Here, x denotes the smallest integer at least x. d e

Lemma 3.5 ([7], Lemma 3). Let G be a graph. Let S1 = a1, a2, . . . , an { } and S2 = b1, b2, . . . , bn be disjoint subsets of V (G) such that the only edges { } between S1 and S2 in G are the edges aibi, where i 1, 2, . . . , n . Then, box(G) n/2 . ∈ { } ≥ d e Theorem 3.6. For a graph G with l universal vertices, the following inequality holds: l box(M(G)) box(G) + . ≥ 2 BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 69

Proof. Let G be a graph and x1, x2, . . . , xl universal vertices of G. Let H be the subgraph of G induced by V (G) x1, x2, . . . , xl . Note that box(H) = box(G) holds. We consider the Mycielski\{ graph M} (G) and its complement M(G). Let Xj = (x1)j, (x2)j,..., (xl)j , the set of vertices in { } V (G)j corresponding to universal vertices of G. Let Dl be the subgraph of M(G) induced by the union of X1 and X2. Note that X1 and X2 are disjoint by their definition. It is not difficult to check that the only edges between X1 and X2 in Dl are the edges (xi)1(xi)2, where i 1, 2, . . . , l . In fact, ∈ { } the vertex (xi)1 X1 is adjacent to all vertices in V (G)2 (xi)2 in M(G) ∈ \{ } and the vertex (xi)2 X2 is adjacent to all vertices in V (G)1 (xi)1 in ∈ \{ } M(G) since xi is a universal vertex of G. We see that box(Dl) l/2 by Lemma 3.5. ≥ d e We prove that d (H, Dl) 2 holds. Let v be a vertex of H and x a M(G) ≥ vertex of Dl. The vertex v is in V (G)1 X1 and the vertex x is in X1 or \ X2. We may represent x as (xi)j, where j 1, 2 . Since xi is a universal ∈ { } vertex of G, the vertex (xi)j is not adjacent to v in M(G). This implies that d (v, x) 2 for a vertex v of H and a vertex x of Dl, that is, M(G) ≥ d (H, Dl) 2. Thus, the inequality M(G) ≥ l box(M(G)) box(H) + box(Dl) box(G) + ≥ ≥ 2 holds by Lemma 3.4. Remark 3.7. We note that the proof of Theorem 3.6 works on the generalized Mycielski graph Mr(G) (see Section 5 for definition), that is, the inequality box(Mr(G)) box(G) + l/2 holds for a graph with l universal ≥ d e vertices. Further observations on box(Mr(G)) appear in Section 5. In the proof of Theorem 3.6, we prove that box(Dl) l/2 by using ≥ d e Lemma 3.5. In fact, note that box(Dl) = l/2 . Any two vertices in X1 d e are not adjacent in M(G) since they are adjacent in M(G). Hence, X1 is independent in Dl. Also note that X2 is a clique in M(G), that is, in Dl by the definition of Mycielski graphs. See the argument behind the proof of Theorem 5 in [7]. If we restrict our attention to the graph G with only one universal vertex or only two universal vertices in the proof of Theorem 3.6, then Lemma 3.5 is superfluous. Note that box(D1) = box(D2) = 1 since D1 is the trivial graph with two vertices and D2 is the path with four vertices. Theorem 3.6 implies that box(M(G)) > box(G) holds for a graph G with universal vertices. Also note that Mycielski’s graph operation can be used to construct graphs with arbitrarily large boxicity (and chromatic number) the same as the join of graphs. At the end of this section, we note that there is a graph G without universal vertices such that the boxicity of the Mycielski graph M(G) is more than that of G. We give a simple example here. Also see Section 6. 70 AKIRA KAMIBEPPU

Example 3.8. Let Pn be a path with n vertices, where n 2. We see that ≥ box(M(Pn)) = 2. We can give a representation of M(Pn) by a family of boxes in Euclidean 2-space. See Figure 3 below, where we write V (Pn)1 = 0 0 0 1, 2, . . . , n and V (Pn)2 = 1 , 2 , . . . , n and for a vertex v V (M(Pn)) = { } { } ∈ z V (Pn)1 V (Pn)2, Bv denotes a box in Euclidean 2-space corresponding { }∪ ∪ to the vertex v. Also note that M(Pn) contains an induced cycle C5.

B2′ B3

B4′ B5

· · ·· Bz ·· B2k 1 −

B(2k)′ B B B2 2k ··· 4

B(2k 1) − ′ . ..

B5′

B3′

B1′

Figure 3. A representation of M(P2k) by a family of boxes in Euclidean 2-space.

4. An upper bound for the boxicity of Mycielski graphs In this section, we give an upper bound for the boxicity of Mycielski graphs. Moreover we calculate the boxicity of Mycielski graphs of some of complete multipartite graphs. First we determine the boxicity of Mycielski graphs of complete graphs.

Lemma 4.1. For a complete graph Kn, the following equalities hold: ( n/2 if n is odd, box(M(Kn)) = d e n/2 + 1 if n is even. d e Proof. Let H0 be the subgraph of M(Kn) induced by V (M(Kn)) z . We − { } have the inequality box(M(Kn)) box(H0) n/2 by Lemma 3.5. ≥ ≥ d e Let V (Kn) = v1, v2, . . . , vn . To see box(M(Kn)) n/2 + 1, we give { } ≤ d e cointerval subgraphs of M(Kn). Let G0 be the subgraph of M(Kn) induced BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 71 by z, (vn)2 V (Kn)1. We deﬁne Gi to be the subgraph of M(Kn) induced { }∪ by (v2i−1)1, (v2i)1 V (Kn)2, where i 1, 2,..., n/2 1 . Moreover, { } ∪ ∈ { d e − } let G be the subgraph of M(Kn) induced by (vn−1)1, (vn)1 V (Kn)2. dn/2e { } ∪ It is easy to see that the family G0,G1,...,G is a cointerval edge { dn/2e} covering of M(Kn), and hence box(M(Kn)) n/2 + 1 holds. ≤ d e If n is odd, the family G0,G1,...,G is a cointerval edge covering { dn/2e−1} of M(Kn), because the edge (vn)1(vn)2 is covered with the graph G0. Hence we have the equality box(M(Kn)) = n/2 . d e If n is even, that is, n = 2k, we show that box(M(K2k)) > k. Suppose to the contrary that M(K2k) can be covered with k cointerval (spanning) subgraphs H1,H2,...,Hk of M(K2k). Let ej = (vj)1(vj)2 for j 1, 2,..., 2k . ∈ { } The graph Hi contains at most two edges in = e1, e2, . . . , e2k since Hi E { } is cointerval. In fact, the graph Hi must contain two edges in . Otherwise E there is a graph H in = H1,H2,...,Hk which contains only one edge in or which containsH no edges{ in . Hence} the family H of k 1 E E H\{ } − cointerval subgraphs of M(K2k) must cover at least 2k 1 edges in , but − E this is impossible. On the other hand, there is a cointerval graph H∗ in H which contains an edge z(v)1, where the vertex v is in V (K2k). We may assume that the graph H∗ contains two edges es and et in . Hence we see E

(vs)2 (vs)1 (vs)2 (vs)1 z z H H ∗ ∗ (vt)2 (vt)1 (vt)2 (vt)1

Figure 4. The subgraph H∗ of M(K2k) containing edges es and et. that V (H∗) (vs)1, (vs)2, (vt)1, (vt)2, z . We note that ⊃ { } (vs)1(vt)1, (vs)1(vt)2, (vt)1(vs)2, z(vs)2, z(vt)2 E(M(K2k)) 6∈ by the deﬁnition of Mycielski’s construction. If v vs, vt , it follows from 6∈ { } Lemma 3.5 that box(H∗) 2 since (v)1(vs)1, (v)1(vt)1 E(M(K2k)), a ≥ 6∈ contradiction. Hence we may assume that v = vs. We reach the four cases on the graph H∗ indicated in Figure 4. These cases imply that box(H∗) ≥ 72 AKIRA KAMIBEPPU

2, which contradicts our assumption that H∗ is cointerval. Thus we have box(M(K2k)) > k. Hence we obtain the equality box(M(Kn)) = n/2 + 1 d e if n is even.

Remark 4.2. We proved that the inequality box(M(Kn)) n/2 +1 holds in the second paragraph of the proof of Lemma 4.1. We can≤ alsod derivee this inequality by using the minimum cardinality of a vertex cover of M(Kn), that is, using the inequality box(M(Kn)) t(M(Kn))/2 + 1. A subset U of the vertex set of a graph G is a vertex≤ cover b of G if forc each e E(G), ∈ there is a vertex u U such that u is in e. Note that t(M(Kn)) = n + 1. ∈ The edge clique cover number of a graph G, denoted by θ(G), is the minimum cardinality of a family of cliques that covers all edges of G. The following theorem gives us an upper bound for the boxicity of Mycielski graphs. Theorem 4.3. For a graph G with l universal vertices, the inequality l box(M(G)) θ(G) + + 1 ≤ 2 holds. If l is zero or odd, we have the inequality l box(M(G)) θ(G) + . ≤ 2

Proof. Let A1,A2,...,A be a family of cliques in G that covers all { θ(G)} edges of G. Let v1, v2, . . . , vl be all isolated vertices of G and write J = v1, v2, . . . , vl . Note that V (G) = A1 A2 A J. We deﬁne { } ∪ ∪ · · · ∪ θ(G) ∪

V (G) (A ) V (G)2 (Ai)2 2 \ i 2 \ independent z complete z · · · · · ·

(Ai)2 (Ai)1 (Ai)2 (Ai)1 complete The subgraph H (E F ) of M(G) H (E F ) i − i ∪ i i − i ∪ i

Intervals for vertices . in V (G) (A ) . 2 \ i 2 The interval for z Intervals for Intervals for vertices in (A ) vertices in (A ) i 1 ··· ··· i 2

Figure 5. The subgraph Hi (Ei Fi) and an interval rep- − ∪ resentation of Hi (Ei Fi). − ∪ BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 73

Hi to be the subgraph of M(G) induced by (Ai)1 V (G)2 z and let ∪ ∪ { } Ei = xy x, y V (G)2 (Ai)2 and Fi = xy x (Ai)1, y V (G)2 (Ai)2 , { | ∈ \ } { | ∈ ∈ \ } where i 1, 2, . . . , θ(G) . We can check that Hi (Ei Fi) is a cointerval graph∈ {(see Figure 5).} Note that the subgraph− of M∪ (G) induced by J1 J2 z is isomorphic to M(Kl). Hence the edge set of the subgraph ∪ ∪ { } of M(G) isomorphic to M(Kl) can be covered with at most l/2 + 1 coin- d e terval subgraphs as in the proof of Lemma 4.1. Let G0 be the subgraph of M(G) induced by z, (vl)2 J1 and Gi the subgraph of M(G) induced by { } ∪ (v2i−1)1, (v2i)1 J2 for i 1, 2,..., l/2 1 . Moreover, let G be { } ∪ ∈ { d e − } dl/2e the subgraph of M(G) induced by (vl−1)1, (vl)1 J2. We can check that { } ∪ θ(G) + l/2 + 1 cointerval subgraphs H1 (E1 F1),...,H (E d e − ∪ θ(G) − θ(G) ∪ Fθ(G)),G0,G1,...,Gdl/2e cover all edges of M(G). Let e be an edge of E(M(G)). If e z = , we see that e V (G)1 = . ∩ { } 6 ∅ ∩ 6 ∅ Hence there is an i 1, 2, . . . , θ(G) such that e E(Hi (Ei Fi)) or ∈ { } ∈ − ∪ e E(G0). If e z = , we have e V (G)1 V (G)2. Hence, if e V (G)2, ∈ ∩{ } ∅ ⊂ ∪ ⊂ in particular, e (Ai)2 = , we see that e E(Hi (Ei Fi)). If e V (G)2 ∩ 6 ∅ ∈ − ∪ ⊂ and e (Ai)2 = for any i, we see that e J2, and hence e E(Gi) for ∩ ∅ ⊂ ∈ i = 0. If e V (G)1 = , we reach the following two cases: 6 ∩ 6 ∅ (i) e V (G)1 or (ii) e V (G)2 = . ⊂ ∩ 6 ∅ In case (i), the edge e is in some (Ai)1 since the family A1,A2,...,A { θ(G)} of cliques covers all edges of G, and hence we have e E(Hi (Ei Fi)). ∈ − ∪ Now we focus on case (ii). Let u be a vertex in V (G) and Cu the union of cliques in A1,A2,...,A containing the vertex u. If u is an isolated { θ(G)} vertex in G, let Cu be the set u . Then we note u1 V (G)1 is never { } ∈ adjacent to vertices in V (G)2 (Cu)2 on M(G) by the deﬁnition of Mycielski graphs. Hence the following two\ subcases occur:

(ii-1) the edge e connects a vertex of (Ai)1 and a vertex of (Ai)2 for some i or (ii-2) the edge e connects a vertex (vi)1 and a vertex (vi)2, where vi J. ∈ Under subcase (ii-1), we notice that e E(Hi (Ei Fi)). Under subcase ∈ − ∪ (ii-2), we see e E(Gdi/2e). These arguments complete the proof of our ﬁrst statement. ∈ If l = 0, the graphs H1 (E1 F1),...,H (E F ) cover all − ∪ θ(G) − θ(G) ∪ θ(G) edges of M(G). If l is odd, H1 (E1 F1),...,H (E F ), − ∪ θ(G) − θ(G) ∪ θ(G) G0,G1,...,Gdl/2e−1 cover all edges of M(G), because the edge (vl)1(vl)2 is covered with the graph G0. Our second statement follows from similar arguments as above. Theorem 3.6 and Theorem 4.3 pretty much narrow the boxicity of My- cielskians of graphs that satisfy the equality box(G) = θ(G). They also determine the boxicity of some Mycielski graphs. 74 AKIRA KAMIBEPPU

Corollary 4.4. For a graph G with l universal vertices that satisﬁes the equality box(G) = θ(G), the inequalities l l box(G) + box(M(G)) box(G) + + 1 2 ≤ ≤ 2 hold. Moreover if l be zero or odd, the equality l box(M(G)) = box(G) + 2 holds. We can give examples of graphs that satisfy box(G) = θ(G). Recall that the boxicity of a complete k-partite graph Kn1,n2,...,nk is the number of ni which is at least 2. If Kn1,n2,...,nk has l universal vertices, we obtain box(Kn ,n ,...,n ) = k l = θ(Kn ,n ,...,n ). Hence we have 1 2 k − 1 2 k l 2k l box(K ) + = − . n1,n2,...,nk 2 2

Corollary 4.5. For a complete k-partite graph Kn1,n2,...,nk with l universal vertices, the inequalities 2k l 2k l − box(M(Kn ,n ,...,n )) min k, − + 1 2 ≤ 1 2 k ≤ 2 hold. In particular, if l is zero or odd, the equality box(M(Kn1,n2,...,nk )) = (2k l)/2 holds. d − e We present other examples of graphs that satisfy box(G) = θ(G). The graph H whose complement is a chain of cliques is a desired one, where neighboring cliques share exactly one vertex and each clique has at least four vertices. Note that the graph H contains a complete multipartite graph K2,2,...,2 as an induced subgraph and the number of its partite sets is equal to that of maximal cliques of the complement H. Moreover if we consider a graph operation that extends a graph G to a new graph Suspn(G), called the n-suspension of G, we can get more examples that we desire. The vertex set of Suspn(G) is the union of V (G) and the set of new vertices x1, x2, . . . , xn . The edge set of Susp (G) is the union { } n of E(G) and the set xiv v V (G), i 1, 2, . . . , n . Here we assume { | ∈ ∈ { }} that n is an integer at least 2. We see that box(Suspn(G)) = box(G) + 1 and θ(Suspn(G)) = θ(G) + 1 for a graph G by Theorem 2.1 and Lemma 3.4. Hence if the graph G satisﬁes box(G) = θ(G), the equality box(Suspn(G)) = θ(Suspn(G)) holds. We note that the family of graphs satisfying box(G) = θ(G) is not narrow at all.

5. Relations between the boxicity of Mycielski graphs and generalized Mycielski graphs In this section, we consider relations between Mycielski graphs and their generalizations in the context of boxicity. BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 75

Let G be a graph and r an integer at least 2. Let V (G)i be a copy of V (G), where i 1, 2, . . . , r . For each vertex v V (G), the symbol ∈ { } ∈ vi denotes the vertex in V (G)i corresponding to v. The generalized My- cielski graph of G, denoted by Mr(G), is the graph whose vertex set is r z ( i=1V (G)i), the disjoint union of the set of an additional new ver- { } ∪ ∪ r+1 tex z and copies V (G)1,...,V (G)r of V (G), and whose edge set is i=1 Ei, where ∪

E1 = u1v1 uv E(G) , { | ∈ } Ei = ui−1vi, vi−1ui uv E(G) for i 2, 3, . . . , r , and { | ∈ } ∈ { } Er+1 = zur u V (G) . { | ∈ } Note that the graph M2(G) is identical to M(G). First, we present a relation between box(Mr(G)) and box(M2(G)) for a bipartite graph G.

Theorem 5.1. The inequality box(Mr(G)) box(M2(G)) + 2 holds for a bipartite graph G and r 2. ≤ ≥ Proof. We partition V (G) into two partite sets V1 and V2. Fix a family Bx { } of boxes in the optimal dimensional space which represents M2(G). Note that Bu Bu = , Bv Bv = , and Bu Bv = for distinct two vertices 1 ∩ 2 ∅ 1 ∩ 2 ∅ 2 ∩ 2 ∅ u and v of G by the definition of M2(G). Moreover we note that uv E(G), ∈ Bu1 Bv1 = , Bu1 Bv2 = , and Bu2 Bv1 = are equivalent each other. ∩ 6 ∅ ∩ 6 ∅ 0 ∩ 6 ∅ First we define the family B of boxes in (box(M2(G)) + 1)-dimensional { vi } space to give a box-representation of the graph Mr(G) z as follows: for each vertex v V (G), − { } ∈ B0 = B 0 , v1 v1 ( × { } B [i 1, i 1/2] if v V , B0 = v2 1 for i 1, 2,..., r/2 , v2i × − − ∈ Bv [ (i 1/2), (i 1)] if v V2, ∈ { b c} 2 × − − − − ∈ and ( B [ (i 1), (i 3/2)] if v V , B0 = v1 1 for i 2, 3,..., r/2 . v2i 1 × − − − − ∈ − Bv [i 3/2, i 1] if v V2, ∈ { d e} 1 × − − ∈ Take a vertex v V (G) and k 1, 2, . . . , r , then consider the adjacency of ∈ ∈ { } 0 the vertex vk of Mr(G) z from the above family Bvi that we defined. It is easy to see that the− { box} B0 does not have intersection{ } with boxes vk 0 corresponding to vertices in v1, v2, . . . , vr vk . We also see that B { }\{ } vk does not have intersection with boxes that correspond to vertices in V (G)k 0 \ vk for k 2, 3, . . . , r . Clearly, the family B represents the { } ∈ { } { v1 }v∈V (G) subgraph of Mr(G) z induced by V (G)1, so we may assume k 2. If the vertex v is− adjacent { } to a vertex u in G, we can check≥ that the box B0 has nonempty intersection only with boxes B0 and B0 for vk uk 1 uk+1 2 k r 1, and only with the box B0 for k =− r in the family ur 1 ≤ ≤ − − B0 ,B0 ,...,B0 . If the vertex v is not adjacent to a vertex u in G, no { u1 u2 ur } 76 AKIRA KAMIBEPPU boxes in the family B0 ,B0 ,...,B0 have nonempty intersection with u1 u2 ur B0 since B B ={ , B B = ,} and B B = hold. Hence the vk u1 v1 u1 v2 u2 v1 0 ∩ ∅ ∩ ∅ ∩ ∅ family Bvi represents the graph Mr(G) z . { } 00 − { } Now, we define the family B of boxes in (box(M2(G))+2)-dimensional { x} space that represents Mr(G) as follows: B00 = B0 0 for i = r, vi vi × { } 6 B00 = B0 [0, 1], vr vr × B00 = B 1 , z × { } where B is a box in (box(M2(G)) + 1)-dimensional space that contains all boxes in B0 . We can check easily that the family B00 represents { vr }v∈V (G) { x} Mr(G), which completes the proof of our theorem.

We believe that the inequality box(Mr(G)) box(M2(G)) + c holds for a graph G and some small constant c in general.≤ The next theorem shows a relation between box(Mr(G)) and box(M3(G)) for a graph G. These results provide further motivation to investigate the boxicity of M2(G) and M3(G).

Theorem 5.2. The inequality box(Mr(G)) box(M3(G)) + 1 holds for a graph G and r 3. ≤ ≥ Proof. Let Bx be a family of boxes in the optimal dimensional space which { } represents M3(G). We note that for distinct two vertices u and v of G,

Bu Bu = ,Bv Bv = , where i, j 1, 2, 3 and i = j, i ∩ j ∅ i ∩ j ∅ ∈ { } 6 Bu Bv = , where i 2, 3 , and i ∩ i ∅ ∈ { } Bu Bv = ,Bv Bu = 1 ∩ 3 ∅ 1 ∩ 3 ∅ hold by the definition of M3(G). In addition we note that uv E(G), ∈ Bu1 Bv1 = , Bu1 Bv2 = , Bu2 Bv1 = , Bu2 Bv3 = , and Bu3 Bv2 = are∩ equivalent6 ∅ each∩ other.6 ∅ By using∩ similar6 ∅ techniques∩ 6 ∅ from the∩ previous6 ∅ 0 theorem, we can present the family B of boxes in (box(M3(G)) + 1)- { x} dimensional space that represents the graph Mr(G) as follows: for each vertex v V (G), define ∈ 0 B = Bv 0 for i 1, 2 , vi i × { } ∈ { } B0 = B [i 2, i 3/2] for i 2, 3,..., r/2 , v2i 1 v3 − × − − ∈ { d e} 0 B = Bv [i 3/2, i 1] for i 2, 3,..., r/2 , v2i 2 × − − ∈ { b c} 0 and for the additional vertex z of Mr(G), ( B r/2 1 if r is even, B0 = × { − } z0 Bz r/2 1/2 if r is odd, × {b c − } where B is a box in (box(M3(G)))-dimensional space that contains all boxes in Bv2 v∈V (G) and z is the additional vertex of M3(G). Note that any pair{ of distinct} two boxes in B0 ,B0 ,...,B0 does not have intersection { v1 v2 vr } BOUNDS FOR THE BOXICITY OF MYCIELSKI GRAPHS 77 for a vertex v of G, and also note that any pair of distinct two boxes in B0 does not have intersection for k 2, 3, . . . , r . vk v∈V (G) { Fix} a vertex v V (G) and k 1, 2, . . . , r∈. { We consider} the adjacency ∈ ∈ { } 0 of the vertex vk of Mr(G). Clearly, the family B represents the { v1 }v∈V (G) subgraph of Mr(G) induced by V (G)1, and hence we may assume k 2. If the vertex v is adjacent to a vertex u in G, we can verify that the box B≥0 has vk nonempty intersection only with boxes B0 and B0 for 2 k r 1, uk 1 uk+1 − ≤ ≤ − and only with B0 for k = r in B0 ,B0 ,...,B0 . If the vertex v is not ur 1 u1 u2 ur − { } adjacent to a vertex u in G, no boxes in the family B0 ,B0 ,...,B0 have u1 u2 ur nonempty intersection with B0 . In addition, the{ box B0 has nonempty} vk vk 0 intersection with Bz if and only if k = r for each v V (G). Hence our 0 0 ∈ arguments guarantee that the family B represents the graph Mr(G). { x} 6. Concluding Remarks: graphs with box(M(G)) > box(G) We proved that the boxicity of the Mycielski graph of a graph G with universal vertices is more than that of G. As examples of complete multipartite graphs without universal vertices, one may expect that the equality box(M(G)) = box(G) holds for a graph G without universal vertices. How- ever, we note that there is a graph G without universal vertices such that box(M(G)) > box(G) holds. For examples, nontrivial interval graphs without universal vertices satisfy this condition. The Mycielski graph of such an interval graph is not interval because it contains a cycle with 5 vertices as an induced subgraph. Another example of a graph without universal vertices that satisfy box(M(G)) > box(G) is a cycle with at least 5 vertices. The author verified in a manuscript that the boxicity of Mycielski graph of a cycle with at least 5 vertices is equal to 3.

Acknowledgment The author would like to thank the anonymous referee for his or her careful reading and suggestions.

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Interdisciplinary Graduate School of Science and Engineering, Shimane University, Matsue, Shimane 690-8504, Japan. E-mail address: [email protected]